Three new constructions of mutually orthogonal latin squares

Let N(n) denote the maximum number of mutually orthogonal Latin squares of order n. It is shown that N(24)≥ 6, N(48) ≥ 7, N(55) ≥ 6. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 218–220, 2000     

Concerning seven and eight mutually orthogonal Latin squares

In this paper, three new direct Mutually Orthogonal Latin Squares (MOLS) constructions are presented for 7 MOLS(24), 7 MOLS(75) and 8 MOLS(36); then using recursive methods, several new constructions for 7 and 8 MOLS are obtained. These reduce the largest value for which 7 MOLS are unknown from 780 to 570, and the largest odd value for which 8 MOLS are unknown from 1935 to 419. © 2003 Wiley Periodicals, Inc.     

Concerning eight mutually orthogonal latin squares

In this article, we provide a direct construction for 8 mutually orthogonal latin squares (MOLS)(48). Using this design together with one of Wilson's recursive constructions produces 8 new MOLS(v) for 88 other values of v. We also mention a few other new sets of 8 and 12 MOLS obtained recursively. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 255–261, 2007     

Five mutually orthogonal Latin squares of order 35

Let N(n) denote the maximum number of mutually orthogonal Latin squares of order n. It is shown that N(35) ≥ 5. © 1996 John Wiley & Sons, Inc.     

Further results on mutually nearly orthogonal Latin squares

Nearly orthogonal Latin squares are useful for conducting experiments eliminating heterogeneity in two directions and using different interventions each at each level. In this paper, some constructions of mutually nearly orthogonal Latin squares are provided. It is proved that there exist 3 MNOLS(2m) if and only if m ≥ 3 and there exist 4 MNOLS(2m) if and only if m ≥ 4 with some possible exceptions.     

The enumeration of cyclic mutually nearly orthogonal Latin squares

In this paper, we study collections of mutually nearly orthogonal Latin squares (MNOLS), which come from a modification of the orthogonality condition for mutually orthogonal Latin squares. In particular, we find the maximum such that there exists a set of cyclic MNOLS of order for , as well as providing a full enumeration of sets and lists of cyclic MNOLS of order under a variety of equivalences with . This resolves in the negative a conjecture that proposed that the maximum for which a set of cyclic MNOLS of order exists is .     

Five mutually orthogonal Latin squares of orders 24 and 40

Let N(n) denote the maximum number of mutually orthogonal Latin squares of order n. It is proved that N(24) and N(40)5.     

Algebraic representation of affine MDS-codes and mutually orthogonal Latin squares

An algebraic representation of affine MDS-codes and of mutually orthogonal Latin squares (MOLS) is given by introducing the term of a partial ternary. The extension respectively lengthening of partial ternaries, MDS-codes and MOLS is discussed.     

A tripling construction for mutually orthogonal symmetric hamiltonian double Latin squares

We provide two new constructions for pairs of mutually orthogonal symmetric hamiltonian double Latin squares. The first is a tripling construction, and the second is derived from known constructions of hamilton cycle decompositions of when is prime.     

New quasi-symmetric designs constructed using mutually orthogonal Latin squares and Hadamard matrices

Using Hadamard matrices and mutually orthogonal Latin squares, we construct two new quasi-symmetric designs, with parameters 2 − (66,30,29) and 2 − (78,36,30). These are the first examples of quasi-symmetric designs with these parameters. The parameters belong to the families 2 − (2u ² − u,u ² − u,u ² − u − 1) and 2 − (2u ² + u,u ²,u ² − u), which are related to Hadamard parameters. The designs correspond to new codes meeting the Grey–Rankin bound.     

More mutually orthogonal latin squares

Wilson's construction for mutually orthogonal Latin squares is generalized. This generalized construction is used to improve known bounds on the function n_r (the largest order for which there do not exist r MOLS). In particular we find n₇?780, n₈?4738, n₉?5842, n₁₀?7222, n₁₁?7478, n₁₂?9286, n₁₃?9476, n₁₅?10632.     

两两正交拉丁方最大数目的新上界

记D（x）是使得TD（x，n）存在的最小的数．本文给出D（x）的一个上界．     

Latin squares with pairwise orthogonal conjugates

Let L¹ denote the set of integers n such that there exists an idempotent Latin square of order n with all of its conjugates distinct and pairwise orthogonal. It is known that L¹ contains all sufficiently large integers. That is, there is a smallest integer n_o such that L¹ contains all integers greater than n_o. However, no upper bound for n_o has been given and the term “sufficiently large” is unspecified. The main purpose of this paper is to establish a concrete upper bound for n_o. In particular it is shown that L¹ contain all integers n>5594, with the possible exception of n=6810.     

Some new conjugate orthogonal Latin squares

We present some new conjugate orthogonal Latin squares which are obtained from a direct method of construction of the starter-adder type. Combining these new constructions with earlier results of K. T. Phelps and the first author, it is shown that a (3, 2, 1)- (or (1, 3, 2)-) conjugate orthogonal Latin square of order v exists for all positive integers v ≠ 2, 6. It is also shown that a (3, 2, 1)- (or (1, 3, 2)-) conjugate orthogonal idempotent Latin square of order v exists for all positive integers v ≠ 2, 3, 6 with one possible exception v = 12, and this result can be used to enlarge the spectrum of a certain class of Mendelsohn designs and provide better results for problems on embedding.     

On the orthogonal Latin squares polytope

Since 1782, when Euler addressed the question of existence of a pair of orthogonal Latin squares (OLS) by stating his famous conjecture, these structures have remained an active area of research. In this paper, we examine the polyhedral aspects of OLS. In particular, we establish the dimension of the OLS polytope, describe all cliques of the underlying intersection graph and categorize them into three classes. Two of these classes are shown to induce facet-defining inequalities of Chvátal rank two. For each such class, we provide a polynomial separation algorithm of the lowest possible complexity.